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The temporal evolution of a quantum system can be characterized by quantumprocess tomography, a complex task that consumes a number of physical resourcesscaling exponentially with the number of subsystems. An alternative approach tothe full reconstruction of a quantum channel allows selecting which coefficientfrom its matrix description to measure, and how accurately, reducing the amountof resources to be polynomial. The possibility of implementing this method isclosely related to the possibility of building a complete set of mutuallyunbiased bases (MUBs) whose existence is known only when the dimension of theHilbert space is the power of a prime number. However, an extension of themethod that uses tensor products of maximal sets of MUBs, has been introducedrecently. Here we explicitly describe how to implement this algorithm toselectively and efficiently estimate any parameter characterizing a quantumprocess in a non-prime power dimension, and we conducted for the first time anexperimental verification of the method in a Hilbert space of dimension $d=6$.That is the small space for which there is no known a complete set of MUBs butit can be decomposed as a tensor product of two other Hilbert spaces ofdimensions $D_1=2$ and $D_2=3$, for which a complete set of MUBs is known. The$6$-dimensional states were codified in the discretized transverse momentum ofthe photon wavefront. The state preparation and detection stages aredynamically programmed with the use of only-phase spatial light modulators, ina versatile experimental setup that allows to implement the algorithm in anyfinite dimension.

Experimental characterization of quantum processes: a selective and efficient method in arbitrary finite dimension